Biot’s modulus

In this analysis, the transient tunnelling process was simulated in a two-dimensional section across the FEBEX tunnel. A coupled HM analysis was conducted using a Biot (1941) model with Young s modulus E = 24.68 GPa, Poisson s ratio v = 0.37, Biot s coefficient b = 1 (Terzaghi assumption), and a Biot s modulus M equal to infinity (the storage phenomena is caused only by skeleton strain). The hydraulic permeability was set to 7xl0 m after model calibration against observed water inflow into the FEBEX tunnel. 

Equation(ll), the variable defines the capillary pressure. The parameter is Biot s modulus related to capillary pressure, which is a function of water saturation degree through the water retention curve... 

Figure 6.1 5 (a) Predicted residual stresses in a sheet of polycarbonate vs. Biot s modulus (from... 

Figure 9.12 Stress reduction factor as a function of non-dimensional time for various values of Biot s modulus, /3.

A parameter variation showed that the HM-induced pressure responses depend on many parameters, including rock-mass deformation modulus, Biot s coupling constants, hydraulic permeability, and the magnitude and orientation of the in situ stress field. The three material parameters affect only the magnitude of the HM-induced pressure response. On the other hand, the magnitude and direction of the in situ stress field are important factors that determine where and when the fluid pressure will increase or decrease. 

The following mechanical properties were assumed for the rock mass density = 2650 kg.m Young s modulus = 35 GPa, Poisson s ratio = 0.22, cohesion = 5 MPa and internal friction angle = 30. For the fracture zones Young s modulus = 3.5 GPa, cohesion = 3 MPa and friction angle = 25°. Biot s hydroelastic coefficient, a, was assumed to be 1 everywhere. In situ stresses were assumed to correspond to mean values for the Canadian and Fennoscandian Shields. These increase with depth, with the maximum principal stress in the horizontal NE-SW direction. 

The AECL team used an in-house MOTIF finite-element code (Guvanasen and Chan 2000), which is based on an extension of the classical poroelastic theory of Biot (1941). This code has undergone extensive verification and validation (Chan et al. 2003). The CTH team employed the commercially available, general-purpose finite-element code ABAQUS/Standard 6.3 (ABAQUS manuals). This code adopts a macroscopic thermodynamic approach. The porous medium is considered as a multiphase material, and an effective stress principle is used to describe its behaviour. ABAQUS allows the value of bulk modulus of the mineral grains as an input parameter. In order to select an appropriate value for this low-permeability, low-porosity rock, the CTH team compared the ABACjus solution with Biot s (1941) analytical solution for ID consolidation in the form presented by Chan et al. 2003). 

The above equations can be rendered dimensionless in terms of Thiele s modulus, Biot number for mass transfer, and nondimensional time and distance, which are defined as... 

S/m and fl/ x = Biot numbers for mass and heat transfer 4 and 4 x = Thiele modulus Le = Lewis number A0 i = dimensionless adiabatic temperature rise t) = effectiveness factor kg =mass transfer coefficient (ms-1) Rp = radius of catalyst pellet (m) Da = effective diffusion coefficient (ms-2) r =rate of reaction (molm-3s-1) C —concentration of reactant (molm-3) ... 

In these parameters s designates some characteristic dimensions of the body for the plate it is the half-thickness, whereas for the cylinders and sphere it is the radius. The Biot number compares the relative magnitudes of surface-convection and internal-conduction resistances to heat transfer. The Fourier modulus compares a characteristic body dimension with an approximate temperature-wave penetration depth for a given time r. 

The effective complex Biot constants H, C, and M are defined in terms of the effective drained modulus of the composite, the effective undrained bulk-modulus K, and the effective Skempton s coefficient B as... 

Finally, Ulm et al. [82] have shown that the concrete and primarily the C-S-H phase, controlling its properties, can be classified as a porous material, complying the rules governing the mechanics of porous bodies. Nowadays the properties of these materials can be examined in the nanometric scale, as it has been shown by Nonat [80] and Plassard [81]. They are elastic bodies to which the Biot modulus and coefficient (between 0.61 and 0.71 for C-S-H) could be applied [82]. 

This result reveals that the external concentration difference is strongly dependent on the internal diffusion process when the internal diffusion resistance is negligible, (f)f Tji approaches zero and so does the concentration difference, whereas this difference increases with increasing s and approaches 4>s/(Bi)m in the asymptotic region of strong diffusion effects (T)iexternal concentration difference is negligible unless the Thiele modulus assumes a value close to that of the Biot number for mass. 

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